Winking Lights

Date: 09/12/98 at 20:19:09
From: Lauren
Subject: Problem solving - winking lights
I have no idea how to do this.
An observer on a boat on Port Phillip Bay at night can see 3 different
shore lights. The red one winks every 6 seconds, the white one every
10 seconds and the green one every 14 seconds. Occasionally all three
lights wink on together.
(a) (i) How often do the red and green lights wink on together?
(ii) How often do the green and white lights wink on together?
(iii) How often do the red and white lights wink on together?
(b) How often do all 3 lights wink on together?
The observer notes that at midnight the red and white lights wink on
together. 6 seconds later the green, and obviously the red, lights
wink on.
(c) When will they all next wink together?
(d) When will this next occur on the hour?
Please help.

Date: 09/12/98 at 22:41:05
From: Doctor Gary
Subject: Re: Problem solving - winking lights
Since we know that there are some occasions on which all the lights
"wink" on together, let's let one of those instants be our starting
point for this part of the question.
Call the time 0 seconds, and keep track of when the lights are going
to "wink" on from then on:
If you remember your "common multiples," you'll know that any two
lights will be on "together" at the lowest common multiple (of seconds)
of their respective intervals.
How often do the red and green lights wink on together?
What will be the least number of seconds that go by after all three
lights were on together that the red light (which goes on every six
seconds), and the green light (which goes on every 14 seconds), will
both be on at the same time?
Will it be at 14 seconds? No, because the red light came on at 6
and 12, and won't come on again until 18.
Will it be at 28? No, because the red light came on at 24, and
won't come on again until 30.
Will it be at 42? Yes, because 42 is divisible by 6 and by 14.
You can use this to solve all the rest of the questions in section a.
How often do all 3 lights wink on together?
What's the lowest common multiple of 6, 10 and 14?
Here is a hint for (c) and (d): Use the answer to (a)(iii), and think
of "red and white" together as something that happens every 30 seconds.
Calling midnight "0", and expressing time as seconds after midnight,
the red and white lights will be on together at 30, 60, 90, 120, and
so on.
The green light comes on at 6. It comes on every 14 seconds, or twice
every 28 seconds. Since 28 is two less than 30, the green light
"catches up" (to the red and white lights) by 2 seconds every two
"winks." So it will take three times two "winks" of the green light to
"catch up" to the other two.
Sure enough, the green light will "wink" at 90, which is 12:01:30 a.m.
When will they all next wink together? When will this next occur on
the hour?
Since they all "wink" together every three-and-a-half minutes, and
they winked together at one-and-a-half minutes after midnight, and
17 times 3.5 is 58.5, and 58.5 plus 1.5 is 60, and there are 60
minutes in an hour, I think you can figure this one out.
- Doctor Gary, The Math Forum
http://mathforum.org/dr.math/